In the past, an apparatus has been proposed which analyzes biological magnetic field sources (one species of physical sources) by disposing plural Superconducting QUantum Interference Device (hereinafter referred to as SQUID) magnetometers in a closed condition relative to the living organism, the SQUID magnetometer employing SQUIDs.
The apparatus performs the following processes using a supercomputer. That is,
a) Scattering m-numbered current elements using random numbers within a search space which is searched by the plural SQUID magnetometers. Input parameters of a current source i are position information P(x,y,z) and a current vector I(X,Y,Z), thereby 6 m-numbered parameters xi,yi,zi,Xi,Yi,Zi (i=1,2, . . . m) are determined using random numbers. PA1 b) Calculating a total estimated error E by an estimated error calculating process which is described later. PA1 c) Repeating the following processing d) to g). PA1 d) Selecting a current element arbitrarily, and evaluating parameters of corresponding current element k and the total estimated error. That is, processings of position information Ps (xs, ys, zs)=Pk (xk, yk, zk), a current vector Is (Xs, Ys, Zs)=Ik (Xk, Yk, Zk), and a total estimated error Es=E are carried out, wherein a suffix k indicates data of a current element k, and a suffix s indicates evaluated data. PA1 e) Varying parameters of the current element k by extremely small quantities using random numbers. That is, when extremely small quantities are supposed as .DELTA.x, .DELTA.y, .DELTA.z, .DELTA.X, .DELTA.Y, .DELTA.Z, processing of Pk (xk, yk, zk)=Pk (xk+.DELTA.x, yk+.DELTA.y, zk+.DELTA.z) Ik (Xk, Yk, Zk)=(Xk+.DELTA.X, Yk+.DELTA.Y, Zk+.DELTA.Z) are carried out. PA1 f) Calculating a total estimated error E by the estimated error calculating process which is described later. PA1 g) Comparing the evaluated total estimated error Es and the total estimated error E, and restoring the information which are evaluated at the processing d) when the total estimated error Es is smaller than the total estimated error E. That is, processing of position information pk (xk, yk, zk)=pk (xs, ys, zs), a current vector Ik (Xk, Yk, Zk)=Is (Xs, Ys, Zs), and a total estimated error E=Es are carried out. PA1 1) Carrying out the following processings 2) and 3) for all measuring points j. PA1 2) Carrying out the following processing 3) for all current elements i. PA1 3) Calculating magnetic field Beij(BXeij,BYeij,BZeij) using the Biot-Savart law, the magnetic field Beij being generated at a measuring point j by a current element i. PA1 4) Calculating the magnetic field Bej using the following equation, the magnetic field Bej being generated at the measuring point j by m-numbered current elements. ##EQU1## PA1 5) Carrying out the following processing 6) for all measuring points j. PA1 6) Calculating the estimated error Ej at each measuring point j using the following equation. EQU Ej=(BXj-BXej).sup.2 +(BYj-Byej).sup.2 +(BZj-BZej).sup.2 PA1 7) Calculating the total estimated error E using the following equation. ##EQU2## PA1 1) Converting a specific object function which is given for analysis, to a form of formula (2). PA1 2) Converting independent variables specifically constituting the object function so as to be activities or output values Ui of neuron models, and determining a conversion rule of the activities or output values Ui, the conversion rule being similar to the formula (1). PA1 3) Determining an input pattern to each Ui, that is the initial value of each Ui, so as to be able to be converged to a pattern finally. PA1 4) Repeating the same information processing using the learning rule obtained at 1) and 2) until an output pattern converges or a function En becomes minimum.
The estimated error calculating process is as follows.
I. Magnetic field at each measuring point j(j=1,2, . . . N) is calculated based upon parameters of each current element. That is,
II. Calculating an estimated error Ej based upon a measured value Bj(BXj,BYj,BZj) at each measuring point j and an estimated value Bej based upon all current elements, and calculating a total estimated error E. That is,
When the magnetic field sources are analyzed using the apparatus mentioned above, correct analysis results seem to be obtained finally because parameters of current elements k are varied by extremely small quantities so as to decrease the total estimated error E.
But, when the initial condition of current elements k are determined as is illustrated in FIG. 22(A), the current elements k are varied to a condition illustrated in FIG. 22(B) when 2,400 times of processing are carried out, and the current elements k are varied to a condition illustrated in FIG. 22(C) when 3,600 times of processing are carried out. Consequently, final solutions cannot be obtained. When FIGS. 22(B) and (C) are compared to one another, condition of the current element k is varied very slightly, therefore a disadvantage arises in that final solution cannot be obtained even when a number of times of processing is increased. Even if 3,600 times of processing are carried out using a supercomputer, it takes about 20 minutes, therefore the apparatus cannot be utilized at all.
The inventor has found causes for the disadvantages. The causes are as follows: When the processing is carried out at once, it is hot assured at all to decrease the total estimated error E, and only processing parts of I.1),2),3) and II.5),6) of the estimated error calculating processings can be processed in parallel, while other processing parts cannot be processed in parallel, therefore increasing the speed of the calculation as a whole cannot be achieved even when a parallel processor system is employed.
In recent years, research of an artificial neural network has developed, and it is considered to apply the artificial neural network to the analysis of the magnetic field sources mentioned above.
The artificial neural network is classified into a hierarchical perceptron (refer to FIG. 23) and a Hopfield model (refer to FIG. 26).
A hierarchical perception is constituted by an input layer for receiving an input pattern, one or more intermediate layers and an output layer for outputting an output pattern, and a plurality of neuron devices constituting each layer are interconnected to one another, as is illustrated in FIG. 23. A back propagation rule is employed as a learning rule of the hierarchical perception. But, when a problem arises when the hierarchical perception becomes complicated, and a large-numbered-layer construction is required, or an increase of a number of neuron devices which constitute one intermediate layer is required. Consequently, a number of neuron devices as a whole becomes great, a number of weighting factors which should be determined by learning becomes extremely great, and an operation load for a converging solution becomes extremely great. Specifically, when learning is carried out for about 50 times for one pattern, an error suddenly decreases so as to settle to a first convergence value, as is illustrated in FIG. 24. When learning is continued, an error may suddenly vary over several times. FIG. 25 illustrates varying of an error for each pattern. In FIG. 25, the combination of the allocation of errors continues (refer to solid lines in FIG. 25) even at a point when the error as a whole (refer to dashed line in FIG. 25) scarcely exists, therefore the trial for decreasing an error as a whole continues. Consequently, when objective accuracy cannot be obtained by a first convergence value, great amount of processing is required to reach the next convergence, and this does not assure a sufficient accuracy will be obtained by the next convergence value.
As is apparent from the foregoing, it is almost impossible to employ a hierarchical perception for analyzing magnetic field sources for practical use because of limitations concerning to learning process of physical phenomena.
A Hopfield model has an arrangement in which each neuron model is interconnected to all other neuron models, and has no classification such as input layer, intermediate layer and output layer of a hierarchical perception, as is illustrated in FIG. 26. And all neuron models can perform functions as an input layer, intermediate layer, and output layer. When a neuron model is supposably a threshold device model, the varying condition of a neuron model i can be modeled by one of following formulae. ##EQU3##
Ui represents an activity or output value of the neuron model i, hi represents a threshold value of the neuron model i, Wij represents a weighting factor, and i is not equal to j.
The function ##EQU4## (.alpha.i is a positive constant, and i is not equal to j in the first paragraph of the above-mentioned formula) which is defined as an evaluation function for determining weighting factors and a learning rule of a neuron model, by Hopfield, decreases by varying of each neuron model when neuron models vary their interior condition based upon formulae (1) in an asynchronous manner. When formula (2) becomes a minimum value or a local minimum value, activities or output values of the neuron models converge.
When analysis is to be carried out using a Hopfield model, the following works 1) to 4) are necessary for preparation. That is,
Among these works, works 1) and 2) are very difficult and it is highly possible not to convert suitably, therefore an extremely great amount of work is needed. As to the work 3), the output pattern may converge or not depending upon a given input pattern. Therefore, it is difficult to determine an input pattern which gives a high convergence, and a different solution may be obtained depending upon a given input pattern. As to the work 4), it takes a fairly long time from varying of the inner state of one of the neuron models to a new stable state settled by propagating an unbalanced state caused by the variation to the entirety of the Hopfield model. Therefore, it may take an extremely long time to obtain the convergence values by performing information processing in an asynchronous manner.
As is apparent from the foregoing, it is almost impossible to employ a Hopfield model for analyzing magnetic field sources for practical use.
While only the application for analysis of magnetic field sources is described in the foregoing, similar disadvantages arise when the system is applied o analysis of other physical sources, such as pressure sources and temperature sources, a regulating rule of the system including the physical sources being able to be expressed by numerical formulae, and linear addition being realized in the system.
The present invention was made to solve the above-mentioned problems. It is an object of the present invention to supply novel methods and apparatus for analyzing physical quantities, the methods and apparatus performing analyzing of a physical source with ease and at a high speed based upon measured values obtained at plural points, and an apparatus for reducing line spectrum noise.